In studying interacting proteins, complementary insights are provided by analyzing both the association model (the stoichiometry and affinity constants of the intermediate and final complexes) and the quaternary structure of the resulting complexes. Many current methods for analyzing protein interactions either give a binary answer to the question of association and no information about quaternary structure or at best provide only part of the complete picture. Presented here is a method to extract both types of information from X-ray or neutron scattering data for a series of equilibrium mixtures containing the initial components at different concentrations. The method determines the association pathway and constants, along with the scattering curves of the individual members of the mixture, so as to best explain the scattering data for the mixtures. The derived curves then enable reconstruction of the intermediate and final complexes. Using simulated solution scattering data for four hetero-oligomeric complexes with different structures, molecular weights and association models, it is demonstrated that this method accurately determines the simulated association model and scattering profiles for the initial components and complexes. Recognizing that experimental mixtures contain static contaminants and nonspecific complexes with the lowest affinities (inter-particle interference) as well as the desired specific complex(es), a new analytical method is also employed to extend this approach to evaluating the association models and scattering curves in the presence of static contaminants, testing both a nonparticipating monomer and a large homo-oligomeric aggregate. It is demonstrated that the method is robust to both random noise and systematic noise from such contaminants, and the treatment of nonspecific complexes is discussed. Finally, it is shown that this method is applicable over a large range of weak association constants typical of specific but transient protein-protein complexes.

Symmetric homo-oligomers represent a majority of proteins, and determining their structures helps elucidate important biological processes, including ion transport, signal transduction, and transcriptional regulation. In order to account for the noise and sparsity in the distance restraints used in Nuclear Magnetic Resonance (NMR) structure determination of cyclic (C n ) symmetric homo-oligomers, and the resulting uncertainty in the determined structures, we develop a Bayesian structural inference approach. In contrast to traditional NMR structure determination methods, which identify a small set of low-energy conformations, the inferential approach characterizes the entire posterior distribution of conformations. Unfortunately, traditional stochastic techniques for inference may under-sample the rugged landscape of the posterior, missing important contributions from high-quality individual conformations and not accounting for the possible aggregate effects on inferred quantities from numerous unsampled conformations. However, by exploiting the geometry of symmetric homo-oligomers, we develop an algorithm that provides provable guarantees for the posterior distribution and the inferred mean atomic coordinates. Using experimental restraints for three proteins, we demonstrate that our approach is able to objectively characterize the structural diversity supported by the data. By simulating spurious and missing restraints, we further demonstrate that our approach is robust, degrading smoothly with noise and sparsity.

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We obtain the lower bound on a variant of the common problem of dimensionality reduction. In this version, the dataset is projected on to a k dimensional subspace with the property that the first k-1 basis vectors are fixed, leaving a single degree of freedom in terms of basis vectors.